When ''P'' is in the upper hemisphere (as in the case of ''P''<sub>1</sub>), ''Q'', the point on the plane, is external to the sphere. When ''P'' is in the lower hemisphere (as in the case of ''P''<sub>2</sub>), ''Q'' falls in the unit circle on the ''x''-''y'' plane.]]

When ''P'' is in the upper hemisphere (as in the case of ''P''<sub>1</sub>), ''Q'', the point on the plane, is external to the sphere. When ''P'' is in the lower hemisphere (as in the case of ''P''<sub>2</sub>), ''Q'' falls in the unit circle on the ''x''-''y'' plane.]]

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The formula for the coordinates of ''Q'', the point on the plane, is:

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'''Rectangular coordinates:''' The formula for the coordinates of ''Q'', the point on the plane, is:

[[Image:Spherical.png|thumb|225px|right|The angles used in the spherical coordinate system. If radius ρ is drawn from the origin to ''P'', then θ is the '''azimuthal''' angle between the positive ''x'' axis and ρ, and ϕ is the '''polar''' angle between the positive ''z'' axis and ρ.]]

[[Image:Spherical.png|thumb|225px|right|The angles used in the spherical coordinate system. If radius ρ is drawn from the origin to ''P'', then θ is the '''azimuthal''' angle between the positive ''x'' axis and ρ, and ϕ is the '''polar''' angle between the positive ''z'' axis and ρ.]]

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Since any <math>P(x, y, z)</math> on the sphere can be written as <math>P(\cos \theta \sin \phi, \sin \theta \sin \phi, \cos \phi)</math> where <math>\phi</math> is the '''polar''' angle formed from the positive ''z'' axis and <math>\theta</math> is the '''azimuthal''' angle formed from the positive ''x'' axis, we may also formulate the [[Change_of_Coordinate_Systems#Three-Dimensional_Coordinates|spherical coordinates]] of the projection:

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'''Projection in terms of spherical coordinates:''' A point on the sphere has rectangular coordinates <math>P(x, y, z)</math>. These rectangular coordinates can be written as <math>P(\cos \theta \sin \phi, \sin \theta \sin \phi, \cos \phi)</math> where <math>\phi</math> is the '''polar''' angle formed from the positive ''z'' axis and <math>\theta</math> is the '''azimuthal''' angle formed from the positive ''x'' axis. In spherical coordinates, we may write equivalently <math>P(1, \theta, \phi)</math> (radius 1 for unit sphere). We can write the coordinates of the projection in terms of these [[Change_of_Coordinate_Systems#Three-Dimensional_Coordinates|spherical coordinates]]:

Contents

Basic Description

Cross section of arbitrary points on a sphere being mapped to points on a plane.

Stereographic projection is a method of mapping the surface of a sphere onto a plane.

Each point on the sphere is associated with a point on the plane. The process for determining the point on the plane is to draw a line from the north pole, letting it pass through both a point on the sphere and a point on the plane. The point on the sphere is mapped to the point on the plane.

The image to the left shows this process for a two-dimensional cross-section of the sphere. In a way, the figure is an example of the unit circle being mapped to the x axis. This page will examine the broader projection of the unit sphere onto the x-y plane.

The main image demonstrates stereographic projection. In this case, the plane is drawn under the sphere instead of cutting through its equator. The coloring demonstrates where regions of the sphere end up when they are mapped to the plane. The projection is still from the top of the sphere, but the bands of color are not centered around the vertical axis, so the projection forms some interesting ellipses on the plane.

The following applet demonstrates how a sphere is projected onto a plane. A sphere with coaxial bands of color is stereographically projected onto a plane in the background. You can rotate the sphere with the mouse, changing the orientation of the colors on the sphere which changes the projection on the plane. The sphere and projection point remain fixed; only the colors are shifted.

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A More Mathematical Explanation

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Coordinates

A stereographic projection maps the points of a sphere onto a plane. Specifical [...]

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Coordinates

A stereographic projection maps the points of a sphere onto a plane. Specifically, let's consider the unit sphere centered at the origin x2 + y2 + z2 = 1 and the x-y plane z = 0. We want to know how to map some point P (x, y, z) on the sphere to some point Q (X, Y, 0) on the plane.

Stereographic projection of two points on a sphere. A line is drawn from the pole through some other point on the sphere and some point on the plane; the point on the sphere is mapped to the point on the plane. All points on the sphere, besides T, can be mapped to the plane in this way.

When P is in the upper hemisphere (as in the case of P1), Q, the point on the plane, is external to the sphere. When P is in the lower hemisphere (as in the case of P2), Q falls in the unit circle on the x-y plane.

Rectangular coordinates: The formula for the coordinates of Q, the point on the plane, is:

Let's restate what is happening by returning to our definition of the sphere. The sphere's pole is at the point T (0, 0, 1). A line can be drawn through some point P (x, y, z) on the sphere and some point Q (X, Y, 0) on the plane. We consider the vectors drawn from T to P and from T to Q. By construction, these two vectors are colinear and parallel:

This inverse formula is derived by recalling the equation for our sphere. By resubstituting, we obtain a quadratic.

The first solution may be discarded because T(0, 0, 1) is the pole of the circle; it is the one point for which stereographic projection is not defined. (Think about why it would not make sense to map T onto the plane. We would have to draw a line from T to T, but The plane tangent to the sphere at T is parallel to the x-y plane onto which we are projecting, so any line tangent to the sphere at T will never pass through the plane.) The second solution is what we expected.

In order to find inverse expressions for x and y, we return once again to the equation for our sphere.

This is certainly a mess to simplify. But in the end we obtain the expected result:

Since x and y are interchangeable for the purpose of these formulas, the same may be repeated for x to obtain:

The angles used in the spherical coordinate system. If radius ρ is drawn from the origin to P, then θ is the azimuthal angle between the positive x axis and ρ, and ϕ is the polar angle between the positive z axis and ρ.

Projection in terms of spherical coordinates: A point on the sphere has rectangular coordinates . These rectangular coordinates can be written as where is the polar angle formed from the positive z axis and is the azimuthal angle formed from the positive x axis. In spherical coordinates, we may write equivalently (radius 1 for unit sphere). We can write the coordinates of the projection in terms of these spherical coordinates:

Properties

Points on the upper hemisphere (z > 0) of the unit sphere are mapped outside of the unit circle on the x-y plane (X2 + Y2 > 1). Points on the lower hemisphere (z < 0) of the unit sphere are mapped inside of the unit circle on the x-y plane (X2 + Y2 < 1).

[Click here to show.]

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We prove this by rewriting the right-hand side of the equation R2 = X2 + Y2 in terms of z using the coordinates in Eq. 1:

For positive z, the numerator is greater than the denominator, so the projection falls outside of the unit circle. For negative z, the denominator is greater than the numerator, so the projection falls inside of the unit circle.

All circles on the surface of the sphere are the intersection of a plane the sphere. The projection of a great circle on the sphere (a circle formed by a plane which contains the center of the sphere) is a line plane. The projection of any other circle on the sphere is a circle on the plane.

The inverse of the stereographic projection. A rectangular coordinate system is projected back onto a sphere, demonstrating that the projection is conformal.

The stereographic projection is conformal, meaning that any angles formed on the surface of the sphere are the same as those projected onto the plane. This means that the projection preserves shape locally; the angles on the sphere and on the plane are the same at the point of intersection.

However, the stereographic projection does not preserve the area of regions on the sphere. This makes sense because the sphere has finite surface area, but is projected onto the whole of the x-y plane (the projection can be inverted and any point on the x-y plane can be mapped to a point on the sphere).

The image to the left shows this property. It is clear that the area of each enclosed region on the rectangular grid is not equal to that of its projection on the sphere. It also appears that the angles are preserved locally. These relationships can be proven.

An important mathematical application of stereographic projections in complex analysis is the Riemann Sphere. In the Riemann sphere, the north pole T of the sphere is the point at infinity (recall that T cannot be projected onto the plane, while the coordinates of the projection become much larger as z approaches 1). The Riemann sphere therefore constitutes the extended complex plane, the union of the complex numbers with infinity.

Why It's Interesting

Stereographic projection in photography. A comparison between a stereographic photograph and the panoramic photograph used to create it. Source.

Photography

Stereographic projections are used in photography to produce impressive, beautiful images.[1]

Such an image as the one to the left is created by taking a panoramic photo. This involves taking a photo, rotating by some fixed angle, and taking another photo. When the photos cover 360°, they may be strung together to form a panoramic photo. It is possible then to map the photo as a texture on a sphere, which can then be projected onto a plane to produce the image on the right of the figure.

Cartography

A difficult task for cartographers has always been to create maps that are as accurate and usable as possible. Various map projections, or projections of earth as an idealized three-dimensional sphere onto a two-dimensional plot, have been used, and each has different merits. Unfortunately, map projections cannot be ideal; they cannot be both conformal and equiareal, but cartographers and geometers have left us many to choose from.

Two polar stereographic projections. To the left, an "extreme" polar map projection maps much of the southern hemisphere (the red line denotes the equation). To the right, a more local polar map projection maps just the northern hemisphere. These are projections from the same point T; the right projection just projects fewer points and so is "zoomed in" and only includes the most accurate portions of the map. Source

The stereographic projection is one such method of mapping. Stereographic projections are chosen because they are conformal and are not distorted much close to the pole.

The general case discussed in the More Mathematical Explanation is a projection onto the plane z = 0. Often, for the purpose of map projections, it makes sense to project the sphere representing the earth onto a plane tangent to a pole of the sphere (for example, z = -1 or z = 1, assuming the unit sphere). An example of how this affects the projection is shown to the right. The formulas for the projection would have to be derived again.

Since the lengths of segments on the sphere are not preserved by the stereographic projection, stereographic map projections can become rather distorted at some points. Particularly, stereographic projections are most accurate around the pole opposite of T (see More Mathematical Explanation). Many stereographic maps are called polar for this reason; they are used to map one hemisphere and are often centered around one of the poles.

For this reason, multiple polar stereographic maps might be necessary to have an accurate picture of the whole world. The two stereographic projections to the left are both polar stereographic map projections, but the first does not seem to represent the whole world very well. The red circle denotes the equator, and the map become very distorted outside of the equator in the southern hemisphere.

Another way to map more of the world using a stereographic projection is with a transverse stereographic map.